Transformation Geometry — Math 331

March 1, 2004

Reflections and Glide Reflections

An isometry is orientation-reversing if and only if relative to Cartesian coordinates it has the form f(x) = Ux + v where U is a reflection matrix, i.e., an orthogonal matrix of determinant −1.

Proposition. If U is a 2 × 2 orthogonal matrix with determinant −1, then the linear transformation σ(x) = Ux is the reflection in a line through the origin, and the following statements hold:

1. 1 − U^{2} = (1 + U)(1 − U) = (1 − U)(1 + U) = 0.

2. Any vector v of the form (1 + U)w for some vector w lies on the axis of σ.

3. Any vector v of the form (1 − U)w for some vector w is perpendicular to the axis of σ.

Proof. It is elementary (see the assignment due February 20) that U has the form

U = ( \begin{matrix}a & b \\ b & −a \end{matrix} )

where a^{2}+b^{2} = 1. Since both U^{−1} = ^{t}U and ^{t}U = U, clearly U^{2} = 1, and, therefore, (1+U)(1−U) = (1−U)(1+U) = 0. If v = (1 + U)w, then (1 − U)v = (1 − U)(1 + U)w = 0, hence, Uv = v, and, therefore, v lies on the axis of σ. If, on the other hand v = (1 − U)w, then by similar reasoning Uv = −v, which characterizes vectors v perpendicular to the axis of σ.

Proposition. Let f = Ux + v be an orientation-reversing isometry, and let

v′ = \frac{1}{2} (1 − U) v and v′′ = \frac{1}{2} (1 + U) v .

Then f is the composition of the isometry σ(x) = Ux + v′, which is a reflection with axis parallel to the axis of the reflection x ↦ Ux, followed by the translation τ(x) = x + v′′ by the vector v′′, which is parallel to the axis of σ.

Proof. Clearly, v′ + v′′ = v, and, therefore, f = τ ∘ σ. Since v′ is perpendicular to the axis of x ↦ Ux, translation by v′ is the composition of the reflections in two lines parallel to the axis of x ↦ Ux, and one of those two lines may be chosen to be the axis of x ↦ Ux and then σ is seen to be the reflection in the other of the two parallel lines. From this follows:

Theorem Let f = Ux + v be an orientation-reversing isometry. Then f is a reflection if and only if Uv = −v and is a glide reflection otherwise.

Exercises due Wednesday, March 3

1. Let f be the affine transformation of the plane defined by

f(x) = ( \begin{matrix}3/5 & 4/5 \\ 4/5 & −3/5 \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) + ( \begin{matrix}1 \\ 2 \end{matrix} ) .

(a) What points x of the plane are “fixed” by f, i.e., satisfy f(x) = x?

(b) What lines L in the plane are “stabilized” by f, i.e., satisfy the condition that f(x) is on L if x is on L?

(c) Find a reflection σ and a translation τ parallel to the axis of σ such that f = τ ∘ σ.

2. Explain briefly why the composition of any four reflections may always be written as the composition of two reflections.

3. Two triangles Δ ABC and Δ A′B′C′ are, by definition, congruent if there is an isometry f for which the vertices of the one triangle are carried to the vertices of the other. If Δ ABC and Δ A′B′C′ are congruent, then how many isometries f of the first with the second are there when

(a) Δ ABC is equilateral.

(b) The sides BC and CA have the same length, which is different from the length of AB.

(c) No two sides of Δ ABC have the same length.